We now introduce some notation. Let \(X^{\alpha }\)  denote the \(\alpha \) -stable process with no negative jump, whose law is characterised by \(\mathbb {E}[\exp (-q X^{\alpha }_{t})] = \exp (t q^{\alpha })\)  for every \(t,q>0\) ; note that \(X^{2}\)  has the law of \(\sqrt{2}\)  times a standard Brownian motion. For every \(t>0\) , let \(d^{\alpha }_{t}\)  denote the density of \(X^{\alpha }_{t}\) .
The classical local limit theorem (see e.g. [1]}, combined with [2]} for the scaling constants) reads
\(\sup _{k \in \mathbb {Z}} \mathopen {}\mathclose {\left|r_{n} \cdot \mathbb {P}\mathopen {}\mathclose {\left(S_{n} = mn+k\right)} - d^{\alpha }_{1}\mathopen {}\mathclose {\left(\frac{k}{r_{n}}\right)}\right|} \mathop {\longrightarrow }^{}_{n \rightarrow \infty }0.\) 
